Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French minerologist Charles-Victor Mauguin. This notation is sometimes called international notation.
The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.[1]
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Nomenclature
Rotational symmetries are denoted by a number n, given by φ = 360 / n, where φ is the angle of rotation. A rotation of 180° would be denoted by the n = 2, and is called a two-fold rotation. By convention these are written down in decreasing order of n, with the largest called the principal axis. Any axes that can be deduced from the others are then removed. Next the additional symmetries of the axes are noted. A bar over n indicates a dihedral symmetry, that is a reflection through a mirror perpendicular to the axis followed by a rotation recovers the original shape. For example the longest diagonal of a cube is 
. Each axis has zero or more mirror planes associated with it (meaning a mirror plane that is perpendicular to the axis or one that is parallel and intersects the axis). An `m' is written after each axis number for each (unique) mirror plane. If one of those mirrors is perpendicular to the axis then a slash is placed between the axis number and the first m. Typically, if given a choice, one shows a mirror plane as a perpendicular.
Thus, for example, a cube has three four-fold axes (through the centre of the faces), four three-fold axes (through the long diagonals), and six two-fold axes (through the centre of diagonally opposite edges). It also has nine mirror planes (three parallel to the faces, and six cutting the faces diagonally). Now note that one of the four-fold axes and one of the three-fold axes will be sufficient to deduce the remaining three- and four-fold axes. Also note that given a mirror plane perpendicular to the four-fold axis allows us to deduce the other two mirror planes that are parallel to the faces. Finally, if we add one of the two-fold axes and its perpendicular mirror plane then all the remaining mirror planes and two-fold axes can be deduced. Hence we notate the cube as 4 / m 2 / m.
Point groups
Point groups exist in both two and three dimensions. They are defined by their symmetry elements, such as the axes of proper and improper rotation and mirror planes. Translational symmetry elements which are present in plane groups and space groups are omitted. Where certain symmetry elements can be deduced, they may be omitted, allowing simplification.
In three dimensions, there are 32 crystallographic point groups:
- 1, 1
- 2, m, 2⁄m
- 222, mm2, mmm
- 4,4, 4⁄m, 422, 4mm, 42m, 4⁄mmm
- 3, 3, 32, 3m, 3m
- 6, 6, 6⁄m, 622, 6mm, 62m, 6⁄mmm
- 23, m3, 432, 43m, m3m
Plane groups
Plane groups can be depicted using the Hermann-Mauguin system. The first letter is either lowercase p or c to represent primitive or centered unit cells. The next number is the rotational symmetry, as given above. The presence of mirror planes are denoted m, while glide reflections are denoted g.
Space groups
Space groups can be defined by combining the point group identifier with the uppercase letters describing the lattice. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group. An example of a space group would be Garnet Ia3d.
Lattice types
These are the Bravais lattices in three dimensions:
- P primitive
- I body centered (after German "Innenzentriert")
- F face centered
- A nodes on A faces only
- B nodes on B faces only
- C nodes on C faces only
Screw axis
The screw axis is noted by a number, n, where the angle of rotation is 
. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (two-fold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (three-fold) rotation followed by a translation of ⅓ of the lattice vector.
The possible screw axis are 21, 31, 41, 42, 61, 62, and 63.
Glide planes
Glide planes are noted by a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.
- nm n-fold screwing with translation by m.
- a, b, or c glide translation along half the lattice vector of this face
- n glide translation along with half a face diagonal
- d glide planes with translation along a quarter of a face diagonal.
- e two with the same glide glide and translation along two (different) half-lattice vectors.
References
- ^ Sands, Donald E.. "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc.. p. 165. ISBN 0-486-67839-3.
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